Callisto

2 years ago

Given that \(y=x^2cosx\) satisfies \(x^2y^{(2)} - 4xy^{(1)} + (x^2 +6)y =0\) , where \( y^{(n)} = \frac{d^ny}{dx^n}\).
(a) Show that when x=0, \((n-2)(n-3)y^{(n)} = -n(n-1)y^{(n-2)}\).
(b) If we use the notation \(\bar y^{(n)} = \frac{d^ny}{dx^n}|_{x=0}\), find \(\bar y^{(2m)}\) and \(\bar y^{(2m+1)}\).
How to do part (b)?